Positive solutions for nonlinear integro-differential equations of mixed type in Banach spaces. (English) Zbl 1470.34210

Sun, Yan

Abstr. Appl. Anal. 2013, Article ID 787038, 6 p. (2013).

Summary: We establish some new existence theorems on the positive solutions for nonlinear integro-differential equations which do not possess any monotone properties in ordered Banach spaces by means of Banach contraction mapping principle and cone theory based on some new comparison results.

MSC:

34K30	Functional-differential equations in abstract spaces
45J05	Integro-ordinary differential equations
45M20	Positive solutions of integral equations

References:

[1]	Guo, D., Extremal solutions of nonlinear Fredholm integral equations in ordered Banach spaces, Northeastern Mathematical Journal, 7, 4, 416-423 (1991) · Zbl 0755.45024
[2]	Deimling, K., Nonlinear Functional Analysis (1985), New York, NY, USA: Springer, New York, NY, USA · Zbl 0559.47040
[3]	Chen, Y. B.; Zhuang, W., The existence of maximal and minimal solution of the nonlinear integro-differential equation in Banach space, Applicable Analysis, 22, 2, 139-147 (1986) · Zbl 0576.45010 · doi:10.1080/00036818608839612
[4]	Du, Y., Fixed points of increasing operators in ordered Banach spaces and applications, Applicable Analysis, 38, 1-2, 1-20 (1990) · Zbl 0671.47054 · doi:10.1080/00036819008839957
[5]	Du, S.; Lakshmikantham, V., Monotone iterative technique for differential equations in a Banach space, Journal of Mathematical Analysis and Applications, 87, 2, 454-459 (1982) · Zbl 0523.34057 · doi:10.1016/0022-247X(82)90134-2
[6]	Ladde, G.; Lakshmikantham, V.; Vatsala, A., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Boston, Mass, USA: Pitman, Boston, Mass, USA · Zbl 0658.35003
[7]	Lakshmikantham, V.; Leela, S., Nonlinear Differential Equations in Abstract Spaces (1981), New York, NY, USA: Pergamon Press, New York, NY, USA · Zbl 0456.34002
[8]	Lakshmikantham, V.; Leela, S.; Vatsala, A., Method of quasi-upper and lower solutions in abstract cones, Nonlinear Analysis: Theory, Methods & Applications, 6, 8, 833-838 (1982) · Zbl 0497.34047 · doi:10.1016/0362-546X(82)90067-0
[9]	Guo, D., Semi-Ordered Method in Nonlinear Analysis, Jinan, China: Shandong Science and Technology Press, Jinan, China
[10]	Su, H.; Liu, L.; Wu, C., Iterative solution for systems of a class of abstract operator equations and applications, Acta Mathematica Sinica, 27, 3, 449-455 (2007) · Zbl 1145.34350
[11]	Zhang, X., Fixed point theorems for a class of nonlinear operators in Banach spaces and applications, Nonlinear Analysis: Theory, Methods & Applications, 69, 2, 536-543 (2008) · Zbl 1168.47041 · doi:10.1016/j.na.2007.05.040
[12]	Liu, L., Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 42, 583-598 (2000) · Zbl 0962.45007 · doi:10.1016/S0362-546X(99)00116-9
[13]	Chen, Y. Z., Existence theorems of coupled fixed points, Journal of Mathematical Analysis and Applications, 154, 1, 142-150 (1991) · Zbl 0719.47041 · doi:10.1016/0022-247X(91)90076-C
[14]	Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Boston, Mass, USA: Academic Press, Boston, Mass, USA · Zbl 0661.47045
[15]	Bhaskar, T. G.; Lakshmikantham, V., Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis: Theory, Methods & Applications, 65, 7, 1379-1393 (2006) · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017
[16]	Krasnosel’skii, M.; Zabreiko, P., Geometrical Methods of Nonlinear Analysis (1984), Berlin, Germany: Springer, Berlin, Germany · Zbl 0546.47030 · doi:10.1007/978-3-642-69409-7
[17]	Heinz, H. P., On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Analysis: Theory, Methods & Applications, 7, 12, 1351-1371 (1983) · Zbl 0528.47046 · doi:10.1016/0362-546X(83)90006-8
[18]	Banaś, J.; Goebel, K., Measures of Noncompactness in Banach Spaces. Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60 (1980), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0441.47056
[19]	Mönch, H., Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Analysis: Theory, Methods & Applications, 4, 5, 985-999 (1980) · Zbl 0462.34041 · doi:10.1016/0362-546X(80)90010-3
[20]	Sun, J.; Liu, L., Iterative method for coupled quasi-solutions of mixed monotone operator equations, Applied Mathematics and Computation, 52, 2-3, 301-308 (1992) · Zbl 0763.65041 · doi:10.1016/0096-3003(92)90084-E